Rival national newspaper editors are expected to present a united front at a Downing Street summit on Tuesday morning, after they were summoned to meet David Cameron and culture secretary Maria Miller for the first time since the Leveson report was published.

Editors are expected to get a stern warning from Cameron, who is due to "pop in" to the meeting, originally called by Miller, that they need to quickly progress plans for a tough new regulatory body for the press, free of any serving editors, or face legislation.

On Monday night, during a Commons debate on the Leveson report, MPs heard that editors have been given six weeks to provide detail on the structure and processes of a new watchdog similar to the one recommended by the appeals court judge, free of serving editors, but not underpinned by statute.

Until the Leveson report came out, editors of several papers including the Daily Mail were arguing for continued self-regulation, if they could levy fines of £1m and introduce a more independent board. However, this plan still included a place on the board for serving editors.

The prime minister has expressed "serious concerns and misgivings" about Lord Justice Leveson's recommendation of legislative underpinning for a new press regulator.

With Labour and the Liberal Democrats united in favour, his own backbench MPs split and phone-hacking victims leading a campaign for full implementation, Cameron is under huge pressure to deliver an alternative to Leveson's statutory underpinning proposal.

An online petition launched by the Hacked Off campaign for tougher press regulation has so far attracted more than 135,000 signatures in favour of statutory underpinning.

Ahead of Tuesday's gathering at No 10, Press Complaints Commission chairman Lord Hunt claimed the support of 120 publishers, representing 2,000 editors, for a new independent regulator.

Hunt said it was not necessary to back this body up with legislation because newspapers could instead sign legally-enforceable membership contracts.

Opening a Commons debate on the Leveson proposals on Monday, Miller told MPs the status quo was not an option in the light of the PCC's failure to prevent abuses by elements of the press.

"Change can either come with the support of the press or – if we are given no option - without it. Be in no doubt that if the industry doesn't respond, the government will," she said.

Action "would include legislation" if industry proposals fall short of Leveson's principles that a new regulator must be truly independent of the industry and government and able to impose big fines on newspapers, she said.

"We will not accept a puppet show with the same people pulling the same strings," Miller added.

Officials at Miller's Department for Culture, Media and Sport are drawing up a draft bill to enact Leveson's recommendations in full – mainly to show that it would be unworkable.

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